Spectral Characterizations of Solvability and Stability for Delay Differential-Algebraic Equations

Abstract

The solvability and stability analyses of linear time invariant systems of delay differential-algebraic equations (DDAEs) are analyzed. The behavior approach is applied to DDAEs in order to establish characterizations of their solvability in terms of spectral conditions. Furthermore, examples are delivered to demonstrate that the eigenvalue-based approach in analyzing the exponential stability of dynamical systems is only valid for a special class of DDAEs, namely, non-advanced. Then, a new concept of weak stability is proposed and studied for DDAEs whose matrix coefficients pairwise commute.

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Acknowledgements

The author would like to thank the anonymous referee for his constructive comments and suggestions that improve the quality of this paper. The author also thanks Stephan Trenn for helpful comments and fruitful discussions on the first topic of this article.

Funding

This research is funded by the Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under the project number 101.01-2017.302.

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Correspondence to Phi Ha.

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Ha, P. Spectral Characterizations of Solvability and Stability for Delay Differential-Algebraic Equations. Acta Math Vietnam 43, 715–735 (2018). https://doi.org/10.1007/s40306-018-0279-7

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Keywords

  • Differential-algebraic equations
  • Time delay
  • Matrix polynomials
  • Commutative
  • Exponential stability
  • Weak stability

Mathematics Subject Classification (2010)

  • 34A09
  • 34A12
  • 65L05
  • 65H10